Shedding Light on Curved Mirrors
Program-Support Notes
In this outstanding program, presenter Spiro Liacos uses clear, real-life examples and superb animations
to explain how and why convex and concave mirrors are so useful. Following a brief recap on reflection
in flat mirrors, Spiro shows how convex mirrors are used in a wide variety of safety applications. He then
describes how concave mirrors produce images, and explains the uses of concave reflectors in things like
headlights, satellite dishes and solar cookers. The program comes with a set of excellent practical
activities and question sheets which keep students engaged in their learning for multiple lessons.
CONTENTS
Part A: Introduction.
Part B: Convex Mirrors: how they form images, where they’re used…
Part C: Concave Mirrors: image formation, uses…
Part D: Concave Reflectors: satellite dishes, torches, solar cookers and more…
Part E: Linear Concave and Linear Convex Mirrors: Lots of fun (park mirrors), but serious too…
Download the Question Sheet.
BONUS FEATURE 1: Parabolic Reflectors: the mathematics of parabolas and a fantastic practical
activity for students…
Once students have seen this section of the video, they can complete the Parabolas worksheet and the
Solar Reflectors prac. See the “Parabolas worksheet” file and the “Practical Activity - Linear Parabolic
Solar Reflectors” file.
BONUS FEATURE 2: Using the Mirror’s Focal Point to Draw Ray Diagrams.
Once students have seen this section, they can complete the “Ray Tracing Activity - Curved Mirrors
Extension Worksheet ”.
The text below is more or less the same as the program’s script.
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Part A: Introduction.
Curved Mirrors. They come in two varieties: convex and concave. These kinds of mirrors are everywhere
and they’ve transformed our
lives. Unlike flat mirrors, which
produce images the same size
as the actual object, curved
mirrors can produce magnified
images and diminished images
(that is images which are
smaller than the actual object). But they can also do more!
But before we go further, a quick recap. In the Shedding Light on Reflection video, we saw how light
rays reflect such that the angle of reflection is equal to the angle of incidence. The angles are always
measured between the light ray and the so-called normal, an imaginary line at right angles to the mirror.
In a flat mirror, light rays coming from an object reflect in such a way that they all appear to be coming
from a specific place behind the mirror. This is where the image is located. In flat mirrors, the image is
the same distance behind the mirror as the object is in front of the mirror. When we look at things in a
mirror, our eyes point towards a location that is somewhere behind the mirror. Even though no light is
coming from behind the mirror, the light reflecting from the mirror makes it look as if something is there.
The image is called a virtual image.
Curved mirrors also reflect light in such a way that an image forms, but because they’re curved, the light
rays obviously don’t reflect off them in the same way they do when they reflect off a flat mirror. So how
do these types of mirrors produce images, and how exactly are they useful? Well, let’s have a look at
convex and concave mirrors in a little more detail.
Part B: Convex Mirrors
Convex mirrors are mirrors which curve outwards.
When parallel light rays strike a flat mirror, the reflected rays are all still
parallel. But when light rays strike a convex mirror, the reflected rays spread
out.
Like flat mirrors, convex mirrors also produce images which
are virtual, upright, front-back inverted, and laterally
inverted. An obvious difference though is that the images are
smaller than the objects are in real life.
The outwards curve of the convex mirror also results in a
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wider so-called field of view.
In the convex mirror, we can see a large part of the landscape because everything is shrunk down in size,
whereas in the flat mirror, we can only see the tops of a few trees.
Another way of thinking about it is, if
you’re looking into a flat mirror, only a
narrow range of light beams can reach you.
Light coming from the sides won’t reflect
from the mirror into your eyes. If you’re
standing in front of a convex mirror
though, because of its curved shape, light
from a wide angle can reach your eyes.
For this reason, convex mirrors are used as safety mirrors wherever you
might need a larger field of view.
Some are used to allow drivers coming out of a driveway to see any
pedestrians on the footpath before the car reaches the footpath. Convex
mirrors are also used in carparks, in hospital corridors, offices, shops,
at train stations, and many, many other places..
Convex mirrors are often used in side-view mirrors on cars, because they allow you to see a wider view
of the road behind you.
In the flat mirror of this minibus, you can see only one
person. In the convex mirror though, you can see six
people. When driving, the convex mirror gives you a
wider view of the road.
Many, if not most, trucks, vans, and buses have, on both
the driver side and the non-driver side, both flat mirrors
and convex mirrors.
We can use ray diagrams to work out why convex mirrors
produce diminished images.
Now because the image is smaller, everything seems to be
further away, but in fact this is just an illusion. As you can
see from the ray diagram, the image of the racket is a
shorter distance behind the mirror than the actual racket is
in front of the mirror. It just appears further away (or
further back) because it’s so much smaller.
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Part C: Concave Mirrors
Concave mirrors are mirrors which curve inwards. The word concave
comes from the Latin word “cavus” which means hollow and it’s also
where the words cave, cavity and carve come from. So it should be
easy to remember which is a concave mirror and which is a convex
mirror.
Unlike convex mirrors which reflect parallel light rays outwards,
concave mirrors reflect parallel light rays inwards towards a focus. The distance of this focal point to the
so-called vertex of the mirror is called the focal length.
Concave mirrors produce two types of images depending
on how far the object is from the mirror.
For objects that are close to the mirror, specifically, when
they are closer than the mirror’s focus point, the image is
magnified and upright.
So-called shaving mirrors are concave mirrors. They
allow you to see a magnified image of your face after
you’ve shaved. Make up mirrors are also concave. The
magnified image allows you to see if everything’s been applied properly.
Dentists sometimes use concave mirrors to see a magnified image of your teeth.
When the object is far from the concave
mirror, specifically, when it’s further
than the focal point, like the trees, the
path, and me, a whole new type of image
is produced. This type of image is called
a real image because you can project it
onto a screen.
The light rays coming from the scene all reflect in such a way that the light is focused on the same side of
the mirror as the object, but the image is upside down, and, in this case diminished, that is smaller than
the object.
Placing a screen of some sort at this position will allow you to see the image on the screen, because
reflected light is illuminating the screen.
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Many astronomical telescopes, especially the bigger ones, use large concave mirrors, and smaller
secondary mirrors, to capture light from distant stars and planets and to then focus it onto cameras which
can then take pictures. Because they use mirrors, these kinds of telescopes are called reflecting telescopes.
Now, not all telescopes use mirrors. Many use lenses. Telescopes that only use lenses to focus light are
called “Refracting telescopes”. Refracting telescopes certainly work very well, but, generally speaking,
the more light you can capture from distant objects like stars and galaxies, the better your photos will be,
and to capture lots of light, it’s much, much easier to build a big mirror than it is to build a big lens.
For this reason, most high-end research in the field of astronomy is carried out by reflecting telescopes.
Part D: Concave Reflectors
A concave reflector has the same shape as a concave mirror, but its main role is to focus light or radio
waves.
This concave satellite dish is designed to reflect all the
invisible radio waves coming from a satellite in space to its
focus point where an antenna is placed.
The concentrated signal from the satellite can then be
picked up by the antenna, and from there, wires take the
signal away to be processed. Satellite Dishes all have
different designs but all of them work in much the same way. When the antenna is placed at a point on the
dish’s principal axis, it’s often called a prime-focus dish or a prime-focus dish antenna.
These types of dish antennas are called offset dishes or offset-dish antennas.
The radio waves come in, reflect off the carefully shaped dish and focus at
the antenna which is offset or to the side of the dish’s principal axis. The
dish itself doesn’t point directly towards the satellite. The advantage of this
arrangement is that the antenna doesn’t block any of the radio waves coming
from space, so it’s a little more efficient. The disadvantage is that it’s much harder to design and make an
offset dish because its shape isn’t symmetrical in every direction. To get
an accurate focal point, the curvature in the vertical direction is different
to the curvature in the horizontal direction.
Concave mirrors can also be used to concentrate sunlight. If the mirror is
facing the sun, the point where the reflected rays focus gets really, really
hot. A magnifying glass, can focus light, too, and you may have used one
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to focus light to a tiny point on a piece of paper. But only a small amount of sunlight is concentrated. A
large mirror can reflect and focus a lot more light than a lens, so it can have a more useful heating effect.
In certain parts of the world where electricity isn’t readily available and people, most often women, have
to carry fire wood long distances to their homes, concave reflectors are being used to concentrate sunlight
to cook food and to kill any germs in the water which is drawn from a well. These solar cookers are
making a huge difference in these communities.
Concave reflectors don’t just focus incoming rays; they can also focus
outgoing rays! Many satellite antennas don’t just receive, they
transmit. The invisible radio waves spread out from the transmitter, hit
the reflector and then reflect back out more or less parallel to each
other in a fairly tight beam.
Transmitting dish antennas are used for example by
telecommunications companies to send signals up to satellites which
then beam the signal back down to another part of the planet. Most satellite dishes you see on people’s
rooves are not transmitting dish antennas.
If the radio waves weren’t concentrated into a beam, they would spread out too much, so high-quality
communication using these radio waves would only be possible over short distances of maybe a few tens
of kilometres or so, or the transmitters would have to be hundreds of times more powerful. The actual
satellites have a wide variety of designs depending on how far they are from the earth’s surface and the
type of signals that they transmit and receive.
Most communication satellites, which allow us to see live broadcasts of, say, sporting events like the
Olympics and other news events, are placed in what’s called a geostationary orbit, where they orbit at
exactly the same rate as the earth turns, that is once every 24 hours. To achieve this orbit, they have to
reach a height of about 36,000km above the earth’s surface. Placing communication satellites in this orbit
means that they always appear to be in the same part of the sky, so you don’t have to keep moving your
dish antenna around to track them.
Of course, the beam from a satellite does spread out a little,
which is a good thing, because that way, back on earth, dish
antennas over a fairly wide area, called the satellite’s
footprint, can then receive the signal. This picture which is
not to scale shows a satellite transmitting to a wide area of
Eastern Asia.
Other satellites, like military spy satellites, and weather satellites orbit only a few hundreds of kilometres
above the earth’s surface, in what’s called Low Earth orbit.
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The International Space Station orbits every 90 minutes at a height of
about 400km above the earth’s surface. It has a number of satellite
dishes. This photo shows an astronaut installing one. Space probes sent
to other planets have them as did the spacecraft that took the astronauts
to the moon. Even the lunar rover had one.
Concave reflectors are also used in many ground-based communication systems, for example to send
information between mobile phone towers or from building to building.
And, TV stations send live broadcasts from their mobile units to the studio using a transmitter placed at
the focal point of a concave reflector.
But concave reflectors aren’t just used for communications.
Torches, or flashlights, also make use of concave reflectors. The torch’s light globe sits at the concave
reflector’s focal point. Any light travelling backwards, ends up shining forwards in a tight beam, while
the light shining forwards spreads out to illuminate the surroundings.
Headlights on cars use concave reflectors, too. The globe is again placed at the focal point of a concave
reflector just like it is in a torch, and the light comes out in a fairly concentrated beam. But headlights go
one step further than torches. To stop forward-moving light from spreading out too much, an additional
reflector is often used to reflect this light back towards the concave reflector, so that even more light then
ends up shining more-or-less directly forward.
The headlight, of course, is designed to allow the light to spread out a little so that it illuminates a fair
amount of the road ahead. The actual lamps themselves aren’t really all that bright, but of course they
seem really bright when all the light is being focused into a tight beam.
Of course, you don’t always need a perfectly focused beam. Downlights, outdoor lights, street lights and
lights in car parks and sports fields all use concave reflectors to shine light into a particular direction, but
they’re designed to allow the light to spread out over a wide area, while ensuring that very little light ends
up shining in a direction where it’s not needed, like into the sky for example.
Bar heaters also make use of concave reflectors, because the infrared light that they produce (or the
radiant heat as it’s often called), also reflects.
Part E: Linear Concave and Linear Convex Mirrors.
Linear convex mirrors and linear concave mirrors are used in fun-house mirrors. Because they curve in
only one direction they can magnify an object in, for example, the vertical direction but not in horizontal
direction.
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Linear concave mirrors, or reflectors are also used in, for example, some types of solar power plants. The
reflectors are used to heat oil to a really high temperature. The hot oil is then used to boil water. The highpressure steam then turns a turbine connected to a generator which produces electricity.
Parabolic Reflectors (Bonus Feature 1)
Concave mirrors and reflectors used in, say, telescopes, or satellite dishes aren’t just any old curved shape
and the curve they trace out isn’t circular. In fact concave reflectors curve in the shape of what we call a
“parabola”.
Only a parabola can focus incoming light beams to a definite focal point. Circular reflectors kind of focus
the light, but not really all that well.
The remainder of this section looks at the mathematics of parabolas and finishes with a practical exercise
where students construct a solar hot water heater.
BONUS FEATURE 2: Using the Mirror’s Focal Point to Draw Ray Diagrams.
Now we’ve already used ray diagrams to work out why a convex
mirror produces a diminished image. Another way to draw a ray
diagram is to use the virtual-focus method. Light rays traveling
towards the convex mirror parallel to what we call the mirror’s
“principal axis” don’t just reflect such that the angles of reflection
equal the angles of incidence. The mirror is shaped so that the light
rays spread out in a way that makes them appear to have come from a specific point behind the mirror.
This point is called the mirror’s “virtual focus”.
The light ray travelling parallel to the principal
axis from the racket head will reflect as if it’s
coming from the virtual focus. The light ray,
striking the exact centre of the mirror, will
reflect downwards at the same angle, since the
angle of incidence equals the angle of
reflection. In this case, the angles are both 28
degrees. By tracing the light back, we can see
where the racquet head will appear. Now since
we conveniently placed the bottom of the handle on the Principal axis, we know that the image of the
handle will also appear on the Principal axis and it will be directly below the racquet head.
Once again, we can see that the image is smaller than the actual object.
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(Since it can be difficult to draw a concave mirror or a convex mirror by hand and know what its focal
point is—the ones drawn in this program were computer generated—we sometimes use a small curve,
either convex or concave and a dotted line to represent a curved mirror and then, having drawn the
principal axis, we can just set whatever focal length we want. Some people find that this is the easiest way
of determining the image’s size and location.)
Now what about the images produced by
concave mirrors? We can also use the
focus method to calculate where the image
will appear, similar to the method we used
for convex mirrors.
A light ray travelling parallel to the
mirror’s principal axis will reflect back
towards the focus of the mirror, so that it
appears to be coming from somewhere
back here. A light ray that hits the vertex
of the mirror will reflect downwards such that the angle of reflection equals the angle of incidence, in this
case 45 degrees. If we trace this reflected light ray back, we can see where the image of the teddy’s head
will form. Once again, since we’ve conveniently placed the teddy’s feet along the principal axis, this is
where the magnified and upright image of the whole teddy bear will appear.
A quick word about magnification!
In everyday use, “magnification” means making something bigger. A magnified image is one which is
larger. However, in scientific use, magnification can be less than 1! An image with a “magnification” of
0.5 is half the size of the object. Many scientists and engineers might prefer to use the word “enlarged”
over “magnified” when the magnification is greater than 1.
The video finishes with a brief explanation of why images in convex mirrors appear distorted.
CREDITS:
Holaniku.png (http://commons.wikimedia.org/wiki/File:Holaniku.png) by Xklaim is licensed under CC BY-SA 3.0.
Parabolic trough solar thermal electric power plant 1.jpg
(http://commons.wikimedia.org/wiki/File:Parabolic_trough_solar_thermal_electric_power_plant_1.jpg)
by kjkolb is licenced under CC BY 2.5.
Smallsketch.jpg (http://commons.wikimedia.org/wiki/File:Smallsketch.jpg) by www.TRECers.net is licensed under cc-by-sa-25.
Photos of solar parabolic mirrors for cooking by Solar Household Energy (http://www.she-inc.org)
and by http://www.solarcooking.org (and Project Sol Suffit): http://www.solarcooking.wikia.com and www.solarcooking.be.
(Used with permission)
(and many more)
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